# Typical Approximation Performance for Maximum Coverage Problem

## TL;DR

This paper analyzes the typical performance of belief propagation, greedy, and linear programming algorithms for maximum coverage on sparse graphs, revealing thresholds and differences in their typical effectiveness through theoretical and numerical analysis.

## Contribution

It provides a detailed analysis of the typical performance thresholds of various approximation algorithms for maximum coverage, highlighting differences not evident in worst-case analysis.

## Key findings

- Belief propagation exhibits two distinct thresholds of replica-symmetry and its breaking.
- In low-density regions, belief propagation outperforms other algorithms in typical performance.
- Numerical simulations confirm theoretical predictions and reveal relationships among algorithms.

## Abstract

This study investigated typical performance of approximation algorithms known as belief propagation, greedy algorithm, and linear-programming relaxation for maximum coverage problems on sparse biregular random graphs. After using the cavity method for a corresponding hard-core lattice--gas model, results show that two distinct thresholds of replica-symmetry and its breaking exist in the typical performance threshold of belief propagation. In the low-density region, the superiority of three algorithms in terms of a typical performance threshold is obtained by some theoretical analyses. Although the greedy algorithm and linear-programming relaxation have the same approximation ratio in worst-case performance, their typical performance thresholds are mutually different, indicating the importance of typical performance. Results of numerical simulations validate the theoretical analyses and imply further mutual relations of approximation algorithms.